Questions: Choose the graph of the logarithmic function. h(x) = log(1/5) x

Solution Steps

To graph the logarithmic function \( h(x) = \log_{\frac{1}{5}} x \), we need to understand its behavior. The base of the logarithm is \(\frac{1}{5}\), which is less than 1, indicating that the function is decreasing. The graph will pass through the point (1, 0) because \(\log_{\frac{1}{5}} 1 = 0\). As \(x\) approaches 0 from the right, \(h(x)\) will approach infinity, and as \(x\) increases, \(h(x)\) will approach negative infinity. We can plot several points to visualize the graph.

The given function is a logarithmic function:

\[ h(x) = \log_{\frac{1}{5}} x \]

This function represents the logarithm of \(x\) with base \(\frac{1}{5}\). The base of the logarithm is a fraction less than 1, which means the function is a decreasing function.

For logarithmic functions of the form \(h(x) = \log_b x\), where \(0 < b < 1\), the graph has the following characteristics:

• The graph is decreasing.
• The x-intercept is at \(x = 1\) because \(\log_b 1 = 0\) for any base \(b\).
• The function is undefined for \(x \leq 0\).
• As \(x\) approaches 0 from the right, \(h(x)\) approaches \(-\infty\).
• As \(x\) approaches \(\infty\), \(h(x)\) approaches \(\infty\).

Based on the analysis, the graph of \(h(x) = \log_{\frac{1}{5}} x\) will:

• Pass through the point \((1, 0)\).
• Decrease as \(x\) increases.
• Approach \(-\infty\) as \(x\) approaches 0 from the right.
• Have a vertical asymptote at \(x = 0\).

Final Answer